Probing the inert doublet model using jet substructure with a multivariate analysis

Probing the inert doublet model using jet substructure with a multivariate analysis

Akanksha Bhardwaj,

Partha Konar ,

Tanumoy Mandal,

and Soumya Sadhukhan

1

Physical Research Laboratory (PRL), Ahmedabad - 380009, Gujarat, India

2

Indian Institute of Technology, Gandhinagar-382424, Gujarat, India

3

Department of Physics and Astronomy, Uppsala University, Box 516, SE-751 20 Uppsala, Sweden

4

Department of Physics and Astrophysics, University of Delhi, Delhi 110 007, India (Received 26 June 2019; published 30 September 2019)

We explore the challenging but phenomenologically interesting hierarchical mass spectrum of the inert doublet model where relatively light dark matter along with much heavier scalar states can fully satisfy the constraints on the relic abundance and also fulfill other theoretical as well as collider and astrophysical bounds. To probe this region of parameter space at the LHC, we propose a signal process that combines up to two large radius boosted jets along with substantial missing transverse momentum. Aided by our intuitive signal selection, we capture a hybrid process where the di-fatjet signal is significantly enhanced by the mono-fatjet contribution with minimal effects on the SM di-fatjet background. Substantiated by the sizable mass difference between the scalars, these boosted jets, originally produced from the hadronic decay of massive vector bosons, still carry the inherent footprint of their root. These features implanted inside the jet substructure can provide additional handles to deal with a large background involving QCD jets. We adopt a multivariate analysis using a boosted decision tree to provide a robust mechanism to explore the hierarchical scenario, which would bring almost the entire available parameter space well within reach of the 14 TeV LHC runs with high luminosity.

DOI:10.1103/PhysRevD.100.055040

I. INTRODUCTION

The Standard Model (SM) of particle physics encapsu- lates our knowledge of fundamental interactions of the particle world with all its glory. Until now, apart from a few minor exceptions, the SM is in perfect agreement with all the high energy collider experiments like the Large Hadron Collider (LHC) experiments at CERN. The reputation of the SM, being a complete theory, gets tarnished when it cannot explain the presence of tiny yet nonzero masses of the neutrinos that are already established in neutrino oscillation experiments. Observations of cosmic microwave background radiation in various experiments unambigu- ously establish that 26% of the energy budget of our Universe is made up of an inert, stable component, termed

“dark matter” (DM). The SM does not contain any particle that can satisfy the observed density of the DM, along with

explaining its other properties. The inert doublet model (IDM) is proposed [1,2] as a minimal extension of the SM that can provide an inert weakly interacting DM candidate, stabilized by the discrete symmetry of the model. The SM is extended with an extra SU ð2Þ

scalar doublet which is odd under a discrete Z

symmetry, and thus stabilizes the lightest neutral scalar of the model to be an ideal DM candidate. The neutrino mass can also be arranged in this setup by introducing lepton portals [3].

Inspection of the dark sector of the IDM, as done in Refs. [4,5], reveals that only small islands of parameter space can satisfy the full relic density of the DM dictated by the WMAP and the Planck results. Only light DM with a mass close to half of the Higgs mass can produce full relic density through the resonant Higgs portal annihilation.

Even so, this requires a large mass difference between the DM with the additional beyond the SM (BSM) scalars in the model. Otherwise, the co-annihilation effects in a degenerate mass spectrum reduce the relic density to underproduce DM. Another part of the parameter space where one can explain the full DM relic density is for the DM mass, m

≳ 550 GeV and even that is possible for an extremely degenerate BSM scalar mass spectrum. Among the various DM scenarios discussed above, heavy DM with a hierarchical spectrum can accommodate only about a few percent of the observed relic density and thus makes this

*

akanksha@prl.res.in

†

konar@prl.res.in

‡

tanumoy.mandal@physics.uu.se

§

physicsoumya@gmail.com

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.

scenario uninteresting to probe at the LHC. One of the earlier collider studies of the IDM is performed in Ref. [6].

The dilepton and trilepton signatures at the LHC originat- ing from the IDM have been investigated in Refs. [7,8].

Although the degenerate heavy DM scenario can provide full relic density, the inertness of the model leads to kinematic suppression in heavy DM production at colliders and therefore, makes the signal very weak. Moreover, detection of the soft decay products from such a com- pressed mass spectrum remains challenging due to poor signal efficiency. This scenario is probed using charged track signatures by the CMS Collaboration [9].

Among the two light DM scenarios in this model, the degenerate BSM scalar mass spectrum can satisfy only about 10% of the observed DM relic density. Nonetheless, this case is probed at the LHC through the mono jet search in Ref. [10]. We motivate our framework with the light DM along with the hierarchical mass spectrum where the full DM relic density is achieved. Although very challenging due to tiny production cross sections of the unstable heavy BSM scalars at the LHC, their significant mass differences with the DM candidate give rise to interesting signal topology characterized by two boosted jets along with large missing transverse energy (MET) from the DM production. This gives us a scope to employ sophisticated multivariate analysis equipped with jet substructure vari- ables to isolate the signal from an overwhelmingly large SM background. The search for BSM scalars for this case is studied in the dijet plus MET channel, in a recent study [11]. All these searches do not exhibit bright discovery potential even with the highest possible LHC luminosity. We look to explore a suitable discovery potential of this scenario in this paper.

To reiterate the scenario under consideration, the heavier BSM scalars (a pseudoscalar A and charged Higgs H

) reside in the mass range ( ∼250–700 GeV) much higher than the small mass window (55 –80 GeV) where the DM candidate can lie. Hence, the hierarchical mass difference is large enough for these heavy scalars to decay dominantly to vector bosons (V ¼ fW; Zg), which in turn are sufficiently boosted and corresponding hadronic deposits at the calo- rimeter behave like large radius fatjets, characterized by the jet radius, R ∼ p

=2m

≳ 0.8. Accompanied with large MET acquired by the undetected pair of DM particles, the presence of these fatjets in the signal brings additional variables like fatjet mass (M

) and subjettiness ( τ

), which carry the characteristics of boosted W=Z decay. These observables are perceived as the ones that can distinguish well between the signal and the background, dominantly coming from the SM V þ jets, as only a tiny fraction of QCD jets mimic as boosted jets. Still, when the over- whelmingly large cross section of the background is pitted against the suppressed signal cross section due to inertness of the model, even a tiny fraction can overshadow the signal to deny a significant discovery potential.

There is a possibility of mono-fatjet signal topology [12,13] with roughly 1 order of magnitude higher cross section than the di-fatjet one. In this case, although we have a more significant cross section, the corresponding back- ground becomes uncontrollably large. Therefore, the mono-fatjet topology alone is not sufficient to achieve discovery significance. The di-fatjet topology, on the other hand, also by itself is not enough to produce discovery significance due to tiny production cross section. In this paper, we propose a hybrid topology where the signal selections are designed aiming the di-fatjet topology but also allow a substantial fraction of the mono-fatjet signal.

In doing this, we not only gain in the signal but at the same time, the enormous mono-fatjet background can also be tamed down.

A probe of the IDM using a cut-based analysis (CBA) in the di-fatjet plus MET channel, has failed to reach discovery significance of 5σ in any of our chosen bench- mark points. From the nature of event distribution profiles, it is observed that the two variables viz. M

and τ

are very powerful to separate a tiny signal from the enormous SM background. The discovery significance in CBA is still elusive even if these jet substructure variables are used to the hilt. A multivariate analysis (MVA) in general performs better than a CBA, if appropriate kinematic variables are used in the analysis. So, a sophisticated MVA involving jet substructure variables quoted above is imperative to achieve better discovery potential in the IDM. With the events selected only after baseline cuts (defined later), the signal is still too tiny compared to the background to train the MVA setup. Therefore, the baseline selection criteria should be accompanied by stronger selection cuts at the baseline level to cut down the large background without harming the signal too much before passing events to MVA.

These cuts should be chosen optimally; otherwise, if they are very similar to the one used in CBA, eventually reduce the predictive power of MVA. Finally, our selection cuts are designed in a way such that it allows signal to consist of two high-p

fatjets. Along with the pure di-fatjet signal, substantial contribution comes from the events with mono- fatjet that mimics the signal. This admixture significantly increases the signal cross section but simultaneously bring in some extra background processes in the picture. We perform an MVA analysis coupled with jet substructure variables to achieve improved signal vs background dis- crimination, which could not be realized in the CBA. It helps us to reach a significantly higher LHC discovery potential in the di-fatjet plus MET channel of the IDM.

This paper is organized as follows. In Sec. II we briefly

discuss the IDM, outlining its scalar sector. Next, in

Sec. III, we invoke all the possible theoretical, collider,

and astrophysical constraints applicable to the IDM, to

ascertain the viability of hierarchical BSM scalar sector

along with the presence of a light DM. Subsequently, in

Sec. IV, we discuss four possible DM scenarios depending

on the DM mass and its mass differences to the other BSM scalars to motivate our choice of benchmark points. To define our analysis setup, we list the possible IDM processes contributing to our signal process, di-fatjet plus MET channel in Sec. V . We also discuss all possible SM backgrounds for this channel. At this point, we present our sample benchmark points covering our region of interest, which is also consistent with all the discussed constraints.

In Sec. VI, we first use the baseline cuts and then introduce two fatjet specific observables and study how these perform to increase the signal vs background ratio and obtain the LHC reach for all the benchmark points. In Sec. VII, we improve our probe using MVA to recast the signal vs background numbers with nonrectangular cuts and there- fore, having better sensitivity for the LHC search. Finally, we summarize our results and conclude in Sec. VIII.

II. INERT DOUBLET MODEL

We first discuss the traditional IDM where one adds an additional SU ð2Þ

complex scalar doublet Φ

apart from the SM Higgs doublet Φ

, which are, respectively, odd and even under a discrete Z

symmetry, i.e., Φ

→ Φ

, Φ

→ −Φ

. The most general scalar potential that respects the electroweak symmetry SU ð2Þ

⊗ Uð1Þ

⊗ Z

of the IDM can be written as [5]

VðΦ

; Φ

Þ ¼ μ

Φ

Φ

þ μ

Φ

Φ

þ λ

ðΦ

Φ

Þ

þ λ

ðΦ

Φ

Þ

þ λ

Φ

Φ

Φ

Φ

þ λ

Φ

Φ

Φ

Φ

þ λ

2 ½ðΦ

Φ

Þ

þ H:c:; ð1Þ where Φ

and Φ

both are hypercharged, Y ¼ þ1, and can be written as

Φ

¼
G

vþhþiG

ffiffi

p2



; Φ

¼
H

HþiA

ffiffi

p2



: ð2Þ

Here h is the SM Higgs with G

; G

being the charged and neutral Goldstone bosons, respectively. The charged scalar H

is present in Φ

, along with the neutral scalars, H, A, respectively, being CP even and CP odd. For the vacuum expectation values (VEVs) of the two doublets, we adopt the notation hΦ

i ¼ v= ffiffiffi

p 2

, hΦ

i ¼ 0, keeping in mind the exact nature of the Z

symmetry. The zero VEV of Φ

is responsible for the inertness of this model. Since all the SM fermions are even under Z

, the new scalar doublet does not couple to the SM fermions and thus having no fermionic interactions. The scalar-gauge boson interactions originate through the kinetic term of the two doublets

L

¼ ðD

Φ

Þ

ðD

Φ

Þ þ ðD

Φ

Þ

ðD

Φ

Þ: ð3Þ

All parameters in the scalar potential are assumed to be real in order to keep the IDM CP invariant.

Here, the electroweak symmetry breaking takes place through the SM doublet Φ

getting a VEV, and after this, the masses of the physical scalars at tree level can be written as

m

¼ 2λ

v

; m

¼ μ

þ 1

2 λ

v

; m

¼ μ

þ 1

2 ðλ

þ λ

þ λ

Þv

¼ m

þ 1

2 ðλ

þ λ

Þv

; m

¼ μ

þ 1

2 ðλ

þ λ

− λ

Þv

¼ m

þ 1

2 ðλ

− λ

Þv

: ð4Þ Here, m

is the SM-like Higgs boson mass, and m

are the masses of the CP-even (odd) scalars from the inert doublet, while m

is the charged scalar mass. Either of the neutral scalars can be the DM candidate in this IDM framework since DM observations cannot probe the CP behavior. For the present analysis, we consider the CP-even scalar H as the DM candidate, which corresponds to negative values of λ

parameter. We define λ

þ λ

þ λ

¼ λ

, which can be either positive or negative. The relations between the λ’s and the scalar masses get modified when the QED corrections are considered for both the scalar masses and scalar potential parameters. As the inert scalars do not couple to the SM quarks, higher-order QCD corrections are negligible for these parameters.

Compared to the SM, only the scalar sector is modified in the IDM. Similar to the SM, λ

and μ

are determined by m

≈ 125 and v ≈ 246 GeV. There are five free parameters in the scalar sector of the IDM viz. λ

, λ

, m

, m

, and m

that are expressed in terms of the five scalar potential parameters, μ

and λ

, as shown in Eq. (4). The new doublet, being inert to the SM fermionic sector, does not introduce any additional new parameters in this setup. In IDM the contribution of the self-coupling parameter λ

is mostly limited to fixing unitarity and stability of the potential. It does not affect the scalar masses and their phenomenology. The Higgs portal coupling λ

to the chosen DM candidate H determines the rate of the DM annihilation through the Higgs and, therefore, is an essential parameter in the DM sector along with the DM mass m

¼ m

. The collider phenomenology of the IDM depends on the scalar masses m

; m

and m

, as the mass differences between them play a significant role in propos- ing the search channels for different scenarios.

III. CONSTRAINTS ON THE INERT DOUBLET MODEL

The IDM parameter space is constrained from various

theoretical as well as experimental considerations. In the

model, we have an extra doublet which brings extra parameters in the scalar potential. Therefore, it is imper- ative to check whether the extended potential is bounded from below, i.e., stable at tree level along with the potential parameters being within the unitary and perturbative regime. With the presence of the extra doublet, oblique parameters should be reexamined with respect to the presence of a light DM and custodial symmetry breaking, respectively. The presence of light scalars can also upset the LEP constraints and the Higgs invisible decay limits. Since DM is present in the model, we should satisfy the observed relic density keeping the DM-nucleon interactions below the DM direct detection reach.

A. Theoretical constraints

The scalar sector is modified in the IDM. We ensure the enlarged potential is stable, i.e., not unbounded from below and the global minimum is a neutral one. We also checked if the potential parameters are perturbative at the tree level along with satisfying unitarity bounds.

Bounded from below. —Theoretical constraints on quartic potential parameters ( λ’s) can arise from restricting the scalar potential in Eq. (3) not to produce large negative numbers for large field values, i.e., V > 0 ∀ Φ

→ ∞.

The mixed quartic terms can be combined to form complete square terms and demanding their coefficients to be positive, leads to the following conditions

:

λ

> 0; λ

> 0; λ

þ 2 ffiffiffiffiffiffiffiffiffi λ

λ

p > 0;

λ

þ λ

þ λ

þ 2 ffiffiffiffiffiffiffiffiffi λ

λ

p > 0: ð5Þ

Because of the introduction of new scalars, there are possibilities of having multiple minima. For the inert vacuum to be the global minimum, we restrict it from being charged by ensuring the condition

λ

þ λ

< 0: ð6Þ

We also ensure that the global minimum is the inert vacuum as opposed to an inertlike one, with the imposition of the condition [15,16]

μ ffiffiffiffiffi

λ

p − μ ffiffiffiffiffi

λ

p > 0: ð7Þ

Perturbativity and unitarity.—We form the S matrix with the amplitudes computed from the 2 → 2 scalar scattering processes taking into account all the other quartic terms in the scalar potential. The eigenvalues of the S matrix turn out to be some combinations of these couplings.

The perturbative unitarity constraints on those eigenvalues are jΛ

j ≤ 8π, where the scattering matrix provides us the combinations as

Λ

¼ λ

 λ

; Λ

¼ λ

 λ

; Λ

¼ λ

þ 2λ

 3λ

; Λ

¼ −λ

− λ

 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ðλ

− λ

Þ

þ λ

q ;

Λ

¼ −3λ

− 3λ

 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 9ðλ

− λ

Þ

þ ð2λ

þ λ

Þ

q ;

Λ

¼ −λ

− λ

 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðλ

− λ

Þ

þ λ

q : ð8Þ

B. Collider constraints

Precision measurements at the LEP and the LHC contributed in pinning down the trace of new physics effects in different forms. The effects of hierarchical heavy BSM scalar mass spectrum and the presence of a lighter DM are under consideration. After the discovery of the Higgs boson, the LHC also measured its properties. Two such measurements, the Higgs decay to γγ and its invisible decay are important to consider in the context of IDM.

Oblique correction constraints. —The oblique parame- ters S, T, and U, proposed by Peskin and Takeuchi [17], are different combinations of the oblique corrections, i.e., radiative corrections to the two-point functions of the SM gauge bosons. The S parameter encodes the running of the neutral gauge boson two-point functions (Π

; Π

; Π

) in the lower energy range, from zero momentum to the Z pole.

Therefore, the S parameter is sensitive to the presence of light particles with masses below m

, which is the case here due to the presence of the light DM. The T parameter, on the other hand, measures the difference between the WW and the ZZ two-point functions, Π

and Π

, at zero momentum. Mass splitting of the scalars inside a SU ð2Þ

doublet breaks the custodial symmetry which modifies T.

In the IDM, the mass splittings between the neutral and the charged scalars are controlled by the T parameter. The experimentally measured values of oblique parameters that we use in our analysis are [18]

S ¼ 0.050.11; T ¼ 0.090.13; U ¼ 0.010.11: ð9Þ h → γγ signal strength constraint.—The signal strength for the h → γγ channel is given by the following ratio [19,20],

R

¼ σ ðpp → hÞ

σðpp → h

Þ × BR ðh → γγÞ

BRðh

→ γγÞ : ð10Þ In the IDM, the Higgs production rate is similar to that of the SM, as it is gluon fusion dominated in both the models.

So, in the IDM, the ratio can be approximated as

1

Alternately, for a scalar potential with many quartic cou-

plings, one can consider formulating the copositive matrices to

guarantee the boundedness of the potential [14].

R

¼ BRðh → γγÞ

BR ðh → γγÞ

: ð11Þ

Combined CMS and ATLAS fit in the diphoton channel provides a 2σ limit on this observable as [21],

R

¼ 1.14

: ð12Þ Presence of a charged Higgs in the h → λλ decay loop can induce a significant shift in this ratio for large values of hH

H

coupling. In the IDM, this coupling depends on λ

, which is also related to the charged Higgs mass and this parameter is constrained from the allowed range of the ratio R

that can deviate from unity.

Constraint from the Higgs invisible decay. —Another constraint from the Higgs data, applicable for the scenario when Higgs can decay to a pair of DM particles with a mass m

< m

=2. The invisible decay width is given by

Γðh → InvisibleÞ ¼ λ

v

64πm

1 − 4m

m



2

: ð13Þ

The latest ATLAS constraint on the invisible Higgs decay is [22]

BR ðh → InvisibleÞ ¼ Γðh → InvisibleÞ

Γðh → InvisibleÞþΓðh → SMÞ < 22%:

In the case for light DM when the Higgs decay to a pair of DM particles is kinematically allowed, this limit can significantly constrain the parameter space of the IDM.

LEP bounds. —A reinterpretation of the neutralino search results at the LEP-II has ruled out the parameter regions [23,24] that satisfy the following three conditions:

m

< 80; m

< 100; and m

− m

> 8 GeV: ð14Þ Reinterpretation of chargino search results at the LEP-II has put a bound [25] on the charged Higgs mass as

m

> 70 GeV: ð15Þ A hierarchical IDM scalar spectrum is not restricted from these constraints. Moreover, due to large mass gap in the spectrum, Z → HA; W

→ HH

; W

→ AH

off- shell decays have a negligible effect on the total width of the W and Z bosons, that are very precisely measured at the LEP experiments.

C. Astrophysical constraints

This model contains a DM candidate, the CP-even scalar in Φ

. Therefore, astrophysical constraints on this model consist of the DM relic density and the direct probe of DM in the Xenon and LUX experiments.

Relic density. —There are unputdownable observational pieces of evidence of the presence of DM in a vast range of length scale, starting from intergalactic rotation curve to the latest Planck experiment data. That suggests the current density of the DM comprises approximately 26% energy budget of the present Universe. The observed abundance of DM is usually represented in terms of density parameter Ω as [26]

Ω

h

¼ 0.1187  0.0017; ð16Þ where the observed Hubble constant is H

¼ 100h km s

Mpc

. The rate of DM annihilation to the SM particles is inversely proportional to the relic of the DM, and therefore constraints are imposed to avoid the overproduction of the relic in the IDM. We compute the DM relic density numerically with MicrOmega [27], implementing the IDM details there.

Direct detection constraints. —Along with the con- straints from the relic abundance measurement in the Planck experiment, there exist strict bounds on the DM- nucleon cross section from the DM direct detection experi- ments like Xenon100 (Xenon1T) [28] and more recently from LUX [29]. For scalar DM considered in this work, the spin independent DM-nucleon scattering cross section mediated by the SM Higgs is given as [1]

σ

¼ λ

f

4π

μ

m

m

m

; ð17Þ where μ ¼ m

m

=ðm

þm

Þ is the DM-nucleon reduced mass and λ

¼ ðλ

þ λ

þ λ

Þ is the quartic coupling involved in the DM-Higgs interaction. A recent estimate of the Higgs-nucleon coupling is f ¼ 0.32 [30], although the full range of allowed values is f ¼ 0.26–0.63 [31].

As shown in Fig. 2 later, the Xenon1T upper bound on the DM-nucleon scattering can put a stringent limit on the allowed λ

values that constrain the Higgs-DM coupling.

IV. POSSIBLE SEARCHES AND BENCHMARKING We explore the DM paradigm inside the IDM, discussing the variation of DM relic density with various model parameters. The relic density dependence on DM mass for both the hierarchical and degenerate nature of the BSM mass spectrum is presented in Fig. 1 for different λ

values.

The nature of the mass spectrum is quantified by two mass differences ΔM

≡ m

− m

and ΔM

≡ m

− m

, which are assumed to be equal ( ΔM) in the plots. Effects of the hierarchical mass spectrum with ΔM ¼ 100 GeV and

2

In the IDM, only the Higgs decay rate to γγ can deviate from

the SM value at the leading order. As that deviation is within the

experimental limit, the Higgs boson here easily satisfies all Higgs

signal data.

the degenerate mass spectrum with ΔM ¼ 1 GeV on DM relic density are depicted in Fig. 1(a) and Fig. 1(b), respectively. We also point out how sign reversal of λ

can alter the relic density dependence on the DM mass.

We roughly divide the DM paradigm of the IDM into four different cases depending on the DM mass and the nature of the mass spectrum, specified by ΔM

; ΔM

. These four cases are showcased in Table I. In each scenario, we discuss the thermal DM relic abundance along with the phenomenological study done to probe the BSM scalars.

Case I. —We first consider a case of light DM with mass, m

≲ 80 GeV together with all other heavy scalars within a narrow mass range. This case is severely constrained from the LEP data which rules out m

< 45 GeV. Precise LEP measurements of the Z width constrains Z → AH decay together with the conditions in Eq. (14). Even for the DM mass above 45 GeV, the degenerate nature of the spectrum ensures that all the inert scalars take part in the

co-annihilation processes and reduce the relic density to somewhat below 10% of the total relic. Instead of both of the mass differences ΔM

tiny, if one of them is taken to be large, only the scalar with smaller ΔM dominantly affect the extent of DM co-annihilation. Sharp dip appears when the DM mass is at m

=2 due to the resonant production peak in the DM annihilation through the Higgs portal.

Furthermore, some additional shallow dips in the relic density are also observed when the WW and the ZZ annihilation modes open up, enhancing the annihilation cross section. In this low mass region, the DM annihilation is contributed dominantly through the Higgs portal, and thus the sign of λ

does not affect the relic density. This scenario is explored at the LHC in the mono-jet signal, as discussed in Ref. [10].

Case II. —Here, we consider the light DM with the hierarchical scalar mass spectrum, i.e., large mass differences (ΔM

) with both of the other heavy scalars.

Because of this large mass difference between H and

(a) (b)

FIG. 1. Variation of relic density Ω

h

shown as a function of dark matter mass m

in the inert doublet model. Band of colors with thick dotted lines considering different λ

values. Corresponding negative values of λ

are shown with thin dotted lines. DM relic abundance strongly depends upon the mass difference ΔM between dark matter candidates from additional BSM scalar masses; left and right plots correspond to the large and small values of it, respectively. One can identify four DM paradigms inside the IDM parameter space, as discussed in Table I.

TABLE I. Illustration of four DM paradigms inside the IDM parameter space, comparing DM mass as well as scalar mass hierarchy. Available DM density as a fraction of the required relic density is also pointed out for these cases. We study the phenomenologically interesting but challenging region marked by case II.

M

ΔM Small Large

Small Case I M

< 80 GeV Case III M

∼ 550 GeV

ΔM ∼ Oð1–10Þ GeV ΔM ∼ Oð1Þ GeV

Relic density ∼ 10% Relic density ∼ 100%

Large Case II M

< 80 GeV Case IV M

∼ 550 GeV

ΔM ∼ Oð100Þ GeV ΔM ∼ Oð10–100Þ GeV

Relic density ∼100% Relic density ∼1%

A=H

, the LEP Z-width measurements do not constrain such a low DM mass. DM annihilates only through the Higgs portal and therefore for tiny λ

, the relic is over- produced. However, the entire relic density can be described at larger λ

values, which are progressively bounded from the DM direct detection data from LUX and Xenon1T.

Contrasting this with the degenerate case as pointed out in case I, here the co-annihilation effect is absent in the annihilation cross section and increases the relic density to produce a full relic in the range m

∼ 53–70 GeV depending on different λ

values. Phenomenologically this parameter range is quite interesting although detection of such very light DM along with much heavier other scalars is challenging at the collider. One has to encounter a very small production cross section along with an extremely large SM background where the signal characteristics are very backgroundlike. The LHC potential of this case is studied in the dijet plus MET channel in Ref. [11]. Here, we take up this scenario for further analysis.

Case III. —If we move towards the heavier DM regime, a degenerate mass spectrum can provide full relic density at around m

∼ 550 GeV.

Exact mass depends strongly on the value of λ

parameter. From a 10% relic for m

∼ 100 GeV, it steadily increases as the HH → WW; ZZ annihilations open up and the cross section decreases with mass. The HHVV coupling turns out to be λ

∼ ð4m

ΔM

=v

þ λ

Þ in the limit DM and other heavy scalars are mass degenerate, i.e., ΔM

≈ ΔM

→ 0. We explored this part of the parameter space earlier with both ΔM

¼ 1 GeV. Even if the DM annihilation rate increases with the DM mass, that increase is strongly suppressed due to tiny mass differences between the different BSM scalars in a nearly degenerate mass spectrum. The DM relic density, along with being inversely dependent on the annihilation cross section, also is directly proportional to the DM mass.

Therefore, the interplay of these two competing effects finally ends up in a gradual increase in the DM relic density. The quartic coupling essentially depends only on λ

in ΔM

→ 0, even then a λ

sign reversal does not affect the DM pair annihilation. This scenario is phe- nomenologically interesting but quite challenging to probe at the LHC. This extremely compressed scenario can be probed at the LHC with identifying the charged track signal of a long-lived charged scalar [9].

Case IV. —In the heavier DM regime with hierarchical mass spectrum where both the mass differences are large, e.g., ΔM

≈ ΔM

∼ 100 GeV, the annihilation cross sec- tion increases with the DM mass. This happens due to rapid increase of the DM-gauge boson quartic couplings with its

mass, i.e., λ

¼ 2ð2m

þ ΔM

ÞΔM

=v

þ λ

, for any general ΔM

, which is a result of the large mass difference between the BSM scalars. Enhancement in the DM annihilation leads to drop of the relic density with increasing m

producing up to a few percents of the full observed value. Here, λ

dependence is mostly over- shadowed by large mass differences and does not affect much. In the case of very distinct choices of ΔM

values, the DM annihilation would be dominated by the scalar having a larger mass difference through this enhanced coupling while the other one would contribute negligibly.

Therefore the DM scenario in case III can be envisaged as a limiting case of case IV.

Among the four DM scenarios in the IDM as described above and also summarized in Table I, two cases have emerged as phenomenologically exciting. Light DM (m

∼ 50–80 GeV) with hierarchical mass spectrum with a substantial mass gap ( ΔM

≳ 100 GeV) with other heavy scalars can provide the full observed DM relic density. On the other side, we get a rather heavy DM (m

∼ 550 GeV) with an extremely degenerate mass spectrum, which can also provide the required relic density. Both the scenarios are challenging to probe, as the heavier BSM scalars are difficult to produce in the inert model and essentially confront with large irreducible SM backgrounds.

Now, we particularly focus on the low DM mass (50 –70 GeV) with hierarchical mass spectrum, i.e., ΔM

; ΔM

≳ 200 GeV for our phenomenological study.

To demonstrate the exact numerical evaluation, in Fig. 2, we explore the m

− λ

parameter plane of the IDM for a light DM with ΔM

¼ 100 GeV applying the constraints from the DM relic density measurements, the DM direct detection experiments and the constraint from the Higgs

FIG. 2. Allowed DM relic abundance in m

− jλ

j parameter space extracted for a set of M

; M

, and λ

values. Blue (red) dots are the points where observed abundance is exactly satisfied as in Eq. (16) for þve (−ve) values of λ

. The shaded area under this curve represents over abundance and is thus excluded. Two other shaded regions at the upper portion of the plot are excluded from invisible decay of Higgs and DM direct detection con- straints from XENON1T, Panda, and LUX, respectively.

3

Recently, this limit is brought down to the DM mass

∼400 GeV, as shown in Ref. [32], by introducing right handed

neutrinos, whose late decay to the DM compensates the under-

produced DM relic density seen previously.

invisible decay. This choice of 100 GeV is representative since major annihilation modes for DM are through the Higgs portal and the parameter space of this plot is equally valid for larger ΔM choices. Blue (red) dots are the points where the observed DM relic abundance is exactly satisfied as in Eq. (16) for þve (−ve) values of λ

. The shaded area under this curve represents DM over abundance and thus is excluded. Two other constraints can come from the invisible decay of the Higgs and the DM direct detection constraints from XENON1T, which are shown in the same plane in two other shaded regions in the upper portion of the plot, respectively. The DM direct detection constraints from LUX (Xenon1T) put stringent upper bound on λ

, for all values of light DM. All other constraints described above, provide weaker bounds in this parameter space [33].

With our understanding of allowed DM mass and λ

parameter in the light DM scenario, we now attempt to comprehend other remaining parameters. To do so, we set these parameters to a particular choice from the allowed region of the relic density plot and then perform a scan over the remaining three parameters comprising of heavy scalar masses (M

; M

) and λ

. One such frame of the allowed parameter space after imposing the theoretical constraints (unitarity, perturbativity, etc.) along with the R

constraint from Secs. III A, III B are shown by the blue scatter plots in Fig. 3. The red dots in the same plot represent the values of M

and M

, which satisfy the oblique parameter con- straints. These constraints, primarily through the T param- eter, force these heavy scalar masses M

and M

to be almost degenerate.

To study the specific low mass DM scenario within the IDM, we choose a set of seven benchmark points (BPs) from the allowed parameter space. These BPs covering heavy scalar mass between 250 to 550 GeV along with the corresponding input DM mass and λ parameter are

summarized in Table II. It is worth noting that the choice of M

and λ

is for theoretical and experimental consistency, but the collider analysis proposed in this paper holds equally good for all the allowed points in Fig. 2.

Radiative correction to the DM-Higgs portal coupling is calculated in Ref. [34], which allows slightly more param- eter space.

V. COLLIDER ANALYSIS

We make use of various publicly available HEP pack- ages for our subsequent collider study aiming for a consistent, reliable detector level analysis. We implement the IDM Lagrangian in

[35] to create the UFO [36]

model files for the event generator

[37]

which is used to generate all signal and background events.

These events are generated at the leading order (LO) and the higher-order corrections are included by multiplying appropriate QCD K factors. We use

[38] parton distribution functions for event generation by setting default dynamical renormalization and factorization scales used in

[39]. Events are passed through

[40] to perform showering and hadronization and matched up to two to four additional jets for different processes using

matching scheme [41,42] with virtuality- ordered Pythia showers to remove the double counting of the matrix element partons with parton showers. The matching parameter,

is appropriately determined for different processes as discussed in Ref. [43]. Detector effects are simulated using

[44] with the default CMS card. Fatjets are reconstructed using the F

J

[45]

package by clustering

tower objects. We employ Cambridge-Achen (CA) [46] algorithm with radius param- eter R ¼ 0.8 for jet clustering. Each fatjet is required to have P

at least 180 GeV. We use the adaptive boosted decision tree (BDT) algorithm in the

framework [47]

for MVA.

A. Signal topology

As discussed in the introduction, the hierarchical mass pattern in the IDM scalar sector, (i.e., M

∼ M

≫ M

) provides us with interesting final states. Once a pair of FIG. 3. The blue scatter plot shows the limits from positivity

and perturbativity constraints in the ðM

; M

Þ plane after fixing the DM mass and λ

for all benchmark points. The red dots represent the allowed parameter space after imposing the con- straints from the oblique parameters S and T. Similar allowed parameter space is found for other benchmark points.

TABLE II. Input parameters λ and the relevant scalar masses for some of the chosen benchmark points satisfying all the con- straints coming from DM, Higgs, theoretical calculations, and low energy experimental data as discussed in the text. All the mass parameters are written in units of GeV. Standard choice of the other two parameters are fixed at M

¼ 53.71 GeV and λ

¼ 5.4 × 10

.

Parameters BP1 BP2 BP3 BP4 BP5 BP6 BP7

M

(GeV) 255.3 304.8 350.3 395.8 446.9 503.3 551.8

M

(GeV) 253.9 302.9 347.4 395.1 442.4 500.7 549.63

λ

1.27 1.07 0.135 0.106 3.10 0.693 0.285

heavy scalars (or one heavy scalar associated with a DM candidate) are produced at the LHC, they eventually decay dominantly producing two (or one) boosted vector bosons, each of which is decaying hadronically and thus producing V jet (J

), where V ¼ fW; Zg. These boosted V jets are always associated with large MET (= E

), an outcome of our inability to detect the DM pair at the detector.

Representative Feynman diagrams of these signal topol- ogies are demonstrated in Fig. 4. Among them, it must already be clear to the readers that the 1J

þ = E

channel alone, although being cross-section-wise bigger than 2J

þ = E

, has less sensitivity at the LHC due to over- whelmingly large SM background. Therefore, we primarily focus on the 2J

þ = E

channel where the large background can be tamed down by employing jet substructure variables in an MVA framework. Our signal is not pure 1J

þ = E

or 2J

þ = E

topologies, rather it is an admixture of both processes. Note that the baseline selections (defined in Sec. VI B) are designed keeping 2J

þ = E

topology in mind. This keeps a large fraction of events from the 1J

þ = E

topology. In doing this we gain in the signal, but at the same time one can avoid extremely large background related to the 1J

þ = E

topology, (i.e., by demanding at least one J

instead of two). Before moving on to the actual analysis, we give some useful details about these two signal topologies.

2J

þ = E

channel. —This final state can arise in the IDM for the aforementioned benchmarks from the follow- ing three different channels:

pp → AH

→ ðZHÞðW

HÞ ≡ 2J

þ = E

; pp → H

H

→ ðW

HÞðW

HÞ ≡ 2J

þ = E

;

pp → AA → ðZHÞðZHÞ ≡ 2J

þ = E

: ð18Þ Here, A and H

decay to ZH and W

H, respectively. As Z and W are originating from a heavy resonance, it is possible that they have sufficient boost to be reconstructed in a large radius jet. We do not distinguish a Z jet or a W jet and call them V jet as we always select fatjets with a broad mass range. A V jet possesses a two prong substructure,

i.e., energy distribution is centered around two subjet axes.

We utilize the N-subjettiness ratio τ

(defined later) to tag V jets.

1J

þ = E

channel. —This final state can arise from the following two different channels:

pp → H

H → ðW

HÞH ≡ 1J

þ = E

;

pp → AH → ðZHÞH ≡ 1J

þ = E

: ð19Þ Extra jets can arise in the final state due to initial state radiation (ISR) and can form another fatjet. So these channels can potentially mimic the 2J

þ = E

final state.

We generate matched samples of this signal with up to two additional jets in the final state. In this topology, only one of the two fatjets has the V-jet-like structure and the other jet originates from the QCD radiation which mimics the fatjet characteristics. We find that the contributions to the 2J

þ = E

final state from the 1J

þ = E

topologies are quite significant and sometimes bigger than the 2J

þ = E

contribution itself after our final selection. This is mainly due to bigger production cross sections of pp → AH; H

H processes and the tail events which satisfy the fatjet criteria of our analysis.

The leading order production cross sections for the signal processes discussed above for different BPs are given in Table III. We have used NLO QCD k factors of 1.27 and 1.50 for the q ¯q and the gg initiated productions for the signal [48].

B. Backgrounds

For our hybrid signal discussed in Sec. I as well as in Sec. VA, major backgrounds come from the following SM processes which we discuss briefly below. All these back- grounds are carefully included in our analysis.

(a) (b)

FIG. 4. Representative parton level diagrams of (a) di-V-jet plus missing energy ( 2J

þ = E

) and (b) mono-V-jet plus missing energy ( 1J

þ = E

).

4

The motivation to choose the 2J

þ = E

channel is that one has larger features than in the case of 1J

þ = E

to handle the enormous background, where 1J

þ = E

also contributes to the signal 2J

þ = E

when the extra QCD jet mimics as a fatjet.

The 1J

þ = E

is explored in the searches in Refs. [12,13].

Vþ jets.—There are the following two types of mono- vector boson processes that contribute dominantly in the background.

(i) Z þ jets: This is the most dominant background in our case. We generate the event samples by simu- lating the inclusive pp → Z þ jets → νν þ jets process matched up to four extra partons. Here, invisible decay of Z gives rise to a large amount of

=

E

and QCD jets mimic as fatjets.

(ii) W þ jets: This process also contributes significantly in the background when W decays leptonically, and the lepton does not satisfy the selection criteria. This is often known as the lost lepton background. The neutrino comes from the W decay and contributes to missing energy and QCD jets mimic as fatjets. We generate the event samples by simulating inclusive pp → W þ jets → l

ν þ jets process matched up to four extra partons.

In order to get statistically significant background events coming from the tail phase space region with large = E

, we apply a hard cut of = E

> 100 GeV at the generation level to generate these background events.

VVþ jets.—Different diboson processes like WZ, WW, and ZZ also mimic the signal and contribute to the SM background. The pp → WZ process contributes most significantly among these three diboson channels when W decays hadronically, and Z decays invisibly. We call this background as W

Z

. Similarly, W

W

, where one W decays hadronically and the other leptonically, and Z

Z

(a hadronic Z and a leptonic Z) can also contribute to the SM backgrounds when leptons remain unidentified. All the diboson processes are generated up to two extra jets with

MLM

matching. In this case, one of the fatjets can come from the hadronic decay of V, and the other can come from the hard partons.

Single top. —Single top production in the SM includes three types of processes viz. top associated with W (i.e., pp → tW process), s-channel single top process (i.e., pp → tb), and t-channel single top process (i.e., pp → tj).

Among these, the associated production tW contributes

significantly in the SM background for our signal topologies.

ttþ jets.—This can be a background for our signal topologies when it decays semileptonically, i.e., one of the top decays leptonically and the other decays hadroni- cally. This background contains b jets. We control this background by applying a b veto. This background always has one V jet. Another fatjet can originate from an untagged b jet or QCD radiation.

Apart from the above background processes, we also calculate the contributions from triboson and QCD multijet processes. However, these contributions are found to be insignificant as compared to the background discussed above, and therefore, we neglect the contribution of these backgrounds in the analysis. The production cross sections with higher order QCD corrections for all the background processes considered in this analysis at the 14 TeV LHC are listed in Table IV .

VI. CUT-BASED ANALYSIS

We perform a CBA to estimate the sensitivity of observing the IDM signatures at the high luminosity LHC runs. It is evident that the signal cross sections are too small compared to the vast SM background. Therefore, one needs sophisticated kinematic observables for the isolation of signal events from the background events.

Our signal processes always include at least a hadronically decaying vector boson that can provide a V-like fatjet.

Therefore, we make use of the jet substructure variables for our purpose.

A. V-jet tagging: Jet substructure observables Jet substructure observables have emerged as a powerful technique to search for new physics signatures at colliders.

In our case, boosted W and Z bosons, originated from the decay of heavy IDM scalars (H

, A), give rise to collimated jets that can form a large radius jet (fatjet). These fatjets have two-prong substructures. We utilize two jet TABLE III. Production cross sections for the signal processes

that contribute to the 1J

þ = E

and 2J

þ = E

final states at the 14 TeV LHC. These numbers are for the pp → xy level before the decay of IDM scalars.

σðpp → xyÞ (fb)

Benchmark points AH

H

H

AH

H

H

AA

BP1 34.54 62.53 12.62 7.96 0.50

BP2 18.71 34.12 6.22 4.23 0.40

BP3 11.43 20.84 3.50 2.59 0.34

BP4 7.11 13.32 2.05 1.70 0.28

BP5 4.63 8.44 1.22 1.13 0.24

BP6 2.84 5.32 0.71 0.76 0.19

BP7 1.95 3.70 0.45 0.56 0.16

TABLE IV. Cross sections for the background processes considered in this analysis at the 14 TeV LHC. These numbers are shown with the QCD correction order provided in brackets.

Background process σ (pb)

V þ jets [49,50] Z þ jets 6.33 × 10

[NNLO]

W þ jets 1.95 × 10

[NLO]

VV þ jets [51] WW þ jets 124.31 [NLO]

WZ þ jets 51.82 [NLO]

ZZ þ jets 17.72 [NLO]

Single top [52] tW 83.1 [N

LO]

tb 248.0 [N

LO]

tj 12.35 [N

LO]

Top pair [53] tt þ jets 988.57 [N

LO]

substructure observables viz. the jet-mass (M

) and N- subjettiness ratio ( τ

). The M

is a viable observable to classify the V jets from the fatjets originated from QCD jets. We calculate the jet mass as M

¼ ð P

i∈J

P

Þ

, where P

are the four-vector of energy hits in the calorimeter. The discrimination power of M

reduces if extra contribution comes from the parton, which does not actually originate from the V decay. This results in broadening of the peak in the M

distributions. Different jet grooming techniques are proposed to remove these softer and wide-angle radiations, such as trimming, pruning, and filtering [54 –57] . We choose pruning for grooming the fatjets.

Pruned jet mass. —We performed the pruning with the standard method as prescribed in Refs. [55,56] to clean the softer and wide-angle emission by rerunning the algorithm and vetoing on such recombinations. At each step of recombination, one calculates the two variable z and ΔR

, where z is defined as z ¼ minðP

; P

Þ=P

and ΔR

is the angular separation between two protojets. If z < z

and ΔR

> R

then the ith and jth protojets are not recombined and the softer one is discarded. Here, z

and R

are parameters of the pruning algorithm. We have taken the default values of R

¼ 0.5 and z

¼ 0.1 as

suggested in Ref. [55]. In Fig. 5, we show the distributions for pruned jet mass for signal (BP3) and the important backgrounds. It is evident from these distributions that the peak around 80 –90 GeV reflect the V-mass peak for the signal whereas for most of the background processes the peaks below 20 GeV reflect the fatjets mimic from a single prong hard QCD jet.

N-subjettiness ratio.—N subjettiness is a jet variable which determines the inclusive jet shape by assuming N subjets in it. It is defined as the angular separation of constituents of a jet with the nearest subjet axis weighted by the P

of the constituents and can be calculated as [58,59]

τ

¼ 1 N

X

i

p

min fΔR

; ΔR

; …; ΔR

g: ð20Þ

Here, i runs over the constituent particles inside the jet and p

is the respective transverse momentum. The normali- zation factor is defined as N

¼ P

i

p

R for a jet of radius R. In Fig. 6, the distribution for the N-subjettiness ratio for signal BP3 and leading background are shown.

The value for τ

is small for fatjets emerging from the signal than the background. The N-subjettiness ratio τ

is FIG. 5. Normalized distributions for invariant mass of the leading fatjet M

(left) and subleading fatjet M

(right) after the baseline selection cuts.

FIG. 6. Normalized distributions for N subjettiness of leading fatjet τ

ðJ

Þ (left) and subleading fatjet τ

ðJ

Þ (right) after the baseline

selection cuts.

close to zero if correctly identify the N-prong structure of the jet.

B. Event selection

We list our baseline selection criteria to select events for further analysis.

Baseline selection criteria.—

(i) Events are selected with missing transverse energy

=

E

> 100 GeV.

(ii) We demand for at least two fatjets of radius parameter R ¼ 0.8 constructed using the CA algo- rithm with fatjet transverse momentum P

ðJÞ >

180 GeV.

(iii) We apply the following lepton veto —events are rejected if they contain a lepton with transverse momentum P

ðlÞ > 10 GeV and pseudorapid- ity jηðlÞj < 2.4.

(iv) We further demand that the azimuthal separation Δϕ between the fatjets and = E

, jΔϕðJ; =E

Þj > 0.2. This minimizes the effect of jet mismeasurement con- tributing to = E

.

After primary selection, we apply the following final selection criteria on events satisfying the baseline selection criteria for final analysis.

Final selection criteria. —

(i) After optimization with signal and background, the minimum = E

requirement is raised from 100 to 200 GeV.

(ii) In order to reduce the huge background coming from the tt þ jets, we apply a b veto with p

-dependent b-tagging efficiency as implemented in

. Here, jets are formed using the anti-k

algorithm with radius parameter R ¼ 0.5.

(iii) We demand that the pruned jet mass of leading and subleading fatjets should be in 65 GeV < M

<

105 GeV to tag J

candidates.

(iv) Further, to discriminate the fatjet J

from the QCD jets, we look for the two-prong nature of the fatjet using N subjettiness and select the events with τ

ðJ

Þ < 0.35 of the unpruned fatjet.

In Table V , we present the cut flow for the signal (BP3) associated with the cut efficiencies and the number of events for an integrated luminosity of 3000 fb

at the TABLE V. After implementing the corresponding cut, the expected number of events and cut efficiency are shown

for signal (BP3) for all possible channels that are contributing to the two 2J

þ MET final state, for an integrated luminosity of 3000 fb

at the 14 TeV LHC.

Signal BP3

Cut AH

H

H

AH

H

H

AA

Baseline þ= E

> 200 GeV 672.03 1608.8 711.62 562.15 64.5

[100%] [100%] [100%] [100%] [100%]

b-veto 474.24 1291.74 474.81 426.85 32.5

[70.70%] [80.28%] [66.66%] [75.95%] [50.49%]

65 GeV < MðJ

Þ; MðJ

Þ < 105 GeV 79.50 274.17 171.83 137.56 4.87 [11.83%] [17.04%] [24.12%] [24.48%] [18.13%]

τ

ðJ

Þ; τ

ðJ

Þ < 0.35 52.44 171.79 128.40 101.98 3.5

[7.88%] [10.67%] [18.02%] [18.13%] [5.18%]

TABLE VI. Cut flow for the SM backgrounds after corresponding cuts are implemented, for an integrated luminosity of 3000 fb

at the 14 TeV LHC.

Background

Cut Z

þ jets W

þ jets VV þ jets Single-top tt þ jets

Baseline þ= E

> 200 GeV 3.22 × 10

4.76 × 10

1.47 × 10

2.06 × 10

3.81 × 10

[100%] [100%] [100%] [100%] [100%]

b-veto 2.69 × 10

4.30 × 10

1.13 × 10

3.63 × 10

4.60 × 10

[83.65%] [9.22%] [75.01%] [16.90%] [12.34%]

65 GeV < MðJ

Þ; MðJ

Þ < 105 GeV 3.80 × 10

1.67 × 10

4.09 × 10

1.96 × 10

1.66 × 10

[1.19%] [0.35%] [2.81%] [0.72%] [0.43%]

τ

ðJ

Þ; τ

ðJ

Þ < 0.35 1.30 × 10

3.79 × 10

1.62 × 10

1.44 × 10

3.84 × 10

[0.41%] [0.07%] [1.03%] [0.56%] [0.10%]

14 TeV LHC. Similarly, Table VI represents the cut flow for the different backgrounds. From these numbers, it is explicit that the τ

and M

are powerful variables to have large background reduction with good signal acceptance.

We can further infer from Table V that in spite of quite low efficiencies of AH and H

H channels to satisfy the 2J

þ = E

criteria, they give comparable contributions to the signal due to its large production cross section.

We compute the statistical signal significance using S ¼ N

= ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

N

þ N

p , where N

and N

represent the remaining number of signal and background events after implementing all the cuts. We show the statistical signifi- cance for different benchmark points in Table VII. The highest significance is found for BP3. We would like to emphasize that even after utilizing the novel techniques of jet substructure this particular region of parameter space is very challenging to probe with high sensitivity at the HL-LHC. In order to optimize our search further, we use MVA with jet substructure variables.

VII. MULTIVARIATE ANALYSIS

In the previous section, we present the reach of our model using a CBA. Although we have not achieved discovery significance of 5σ in any of our benchmark points, we see that the two variables viz. M

and τ

are very powerful to separate the tiny signal from the large SM background. In this section, we use a sophisticated MVA to achieve better sensitivity than a CBA. We would like to discuss two important points here. First, we have observed that MVA does not perform well if we use events selected just with the baseline cuts since the signal is too tiny compared to the overwhelmingly large background.

Therefore, we need to apply, in addition to the baseline selection cuts, the following strong cut on the hardest fatjet mass, M

> 40 GeV and b veto on jets to further trim

down the large background before passing events to MVA.

These cuts are very effective to drastically reduce the background but not the signal and are optimally chosen such that it is not too close or too relaxed compared to the cuts used in CBA. If the extra strong cuts for MVA are too close to the cuts applied for the CBA, MVA will not give us an improved sensitivity. On the other hand, if they are too relaxed, the performance of MVA will degrade as the background will become too large. Although we select events with two high-p

fatjets, we only demand the jet mass of the leading-p

fatjet is greater than 40 GeV. This will pass a large fraction of mono-fatjet signal events along with the di-fatjet. Therefore, on the one hand, this will increase the signal. However, on the other hand, this will also increase the background.

In Table VIII, we show the number of signal (1J

and 2J

categories) and background events at the 14 TeV LHC with 3000 fb

integrated luminosity. Observe that although we demand two fatjets in our selection, the number of 1J

events that contribute to the signal are always bigger than the 2J

contributions for all BPs. This is due to the fact that cross sections for 1J

topologies are much bigger than the 2J

topologies and also a significant fraction of 1J

events pass the selection cuts. Therefore, it is necessary to design hybrid selection cuts, stricter than 1J

but looser than 2J

, where both 1J

and 2J

topologies contribute. Our selection cuts are, therefore, optimally designed to achieve better sensitivity.

For our MVA, we use the adaptive BDT algorithm. We obtain two statistically independent event samples for the signal as well as for the background and split the dataset randomly 50% for testing and the rest for training purposes for both the signal and background. Note that there are multiple processes that are contributing to the signal and similarly for the background. In MVA, we construct the signal classes by combining both the 1J

and 2J

topologies that pass our MVA selection criteria. These different signal samples are separately generated at LO and then mixed according to their proper weights to obtain the kinematical distributions for the combined signal.

Similarly, all different background samples are mixed to obtain similar distributions for the background class.

The final set of variables that are used in the MVA are decided from a larger set of kinematic variables by looking TABLE VII. Statistical significance of the signal for different

benchmark points in di-fatjet + = E

analysis for an integrated luminosity of 3000 fb

at the 14 TeV LHC.

Benchmarks BP1 BP2 BP3 BP4 BP5 BP6 BP7

Significance 1.9 2.9 3.2 2.9 1.9 1.6 1.1

TABLE VIII. Number of signal and background events at the 14 TeV LHC with 3000 fb

integrated luminosity.

These numbers are obtained by applying M

> 40 GeV and b-jet veto in addition to the baseline cuts defined in the text.

Topology BP1 BP2 BP3 BP4 BP5 BP6 BP7

1J

1668 2025 2023 1472 1334 1190 920

2J

601 1112 1572 1254 979 948 608

Z W t tt WZ ZZ WW Total

3.15 × 10

1.43 × 10

1.6 × 10

1.6 × 10

1.76 × 10

2.97 × 10

1.21 × 10

5.1 × 10

at their power of discrimination between signal and back- ground classes. Four substructure variables for two fatjets, i.e., M

and τ

ðJ

Þ has already proved to be very important discriminators in our CBA. Stronger transverse momenta cut for such jets are favorable to retain the correct classification of these variables. We already required reasonably high P

criteria for both such jets. However, to construct the hybrid selection cuts P

ðJ

Þ can still take a

role in determining the purity of the hardest fatjet J

. We also include relative separation between these fatjets ΔRðJ

; J

Þ and the azimuthal angle separation between the leading fatjet from the missing transverse momentum direction ΔϕðJ

; = E

Þ. The scale of new physics is relatively high, and that is typically captured by some of the topology independent inclusive variables like H

, = H

, = E

, etc. We utilize the global inclusive variable ffiffiffiffiffiffiffiffiffi

ˆS

p proposed to

FIG. 7. Normalized distributions of the input variables at the LHC ( p ffiffiffi s

¼ 14 TeV) used in the MVA for the signal (blue) and the

background (red). Signal distributions are obtained for BP3 including 1J

and 2J

topologies and the background includes all the

dominant backgrounds discussed in Sec. V B.